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There is a well known essay by Dray and Manogue which argues that differentials should be brought back into freshman calculus, and that we shouldn't worry too much about choosing a specific way of formalizing them, or about giving students a formal answer to the question of "what are differentials?"

I find myself in almost complete agreement with Dray and Manogue, and I feel especially that differentials are the natural way to approach related rates and implicit differentiation. Since differentials are used universally in the sciences and engineering, I think freshman calc students should be exposed to them. I would like to throw in just a taste of differentials when doing related rates and implicit differentiation, without going whole hog and using them throughout the course.

But I will admit to quite a bit of apprehension about actually going ahead with such an approach. I suspect that, for example, I would get some push-back from colleagues who believe that differentials are inherently inconsistent; that students would go to tutors for help, and that the tutors wouldn't understand the approach; and that the kind of casualness about foundational aspects advocated by Dray and Manogue would be interpreted as carelessness or incompetence on my part.

Has anyone implemented anything like what Dray and Manogue advocate, and if so, what were their practical experiences like?

Related: A calculus book that uses differentials? [I was surprised to find that Keisler's treatment of implicit differentiation looks exactly like a standard one, without differentials or infinitesimals.]

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    $\begingroup$ I for one was thoroughly confused by the (informal) infinitesimals rampant in engineering. .. $\endgroup$ – vonbrand May 11 '14 at 0:49
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    $\begingroup$ Agreed, I always was confused by them too. $\endgroup$ – DumpsterDoofus May 11 '14 at 2:17
  • $\begingroup$ @vonbrand: What did you find confusing? Did you eventually overcome your confusion? Would different freshman calc preparation have helped you to avoid the confusion? $\endgroup$ – Ben Crowell May 11 '14 at 4:15
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    $\begingroup$ A good question. But how to define "differential" in an intellectually honest (i.e. rigorous) way? $\endgroup$ – Jyrki Lahtonen May 12 '14 at 6:58
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    $\begingroup$ This is all very interesting! I wrote my Bachelor's Thesis about Leibniz's development of differentials because my pre-university teacher had promised to tell us a little about what they meant, but he never did. BTW I am Danish. We also never defined limits with $\varepsilon,\delta$ before I got to university level. It was all $\Delta x$ approaches this and that. Very disturbing, I think! $\endgroup$ – String May 12 '14 at 7:47
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There is a hybrid approach which is possible:

  1. teach $\frac{df}{dx}$ as traditionally is done. Motivate geometrically by secant lines and physically my instantaneous velocity. Emphasize that $x \mapsto \frac{df}{dx}$ is a new function defined pointwise by the derivative.
  2. introduce differentials as they naturally appear in related rates problems. For example, if $A = lw$ and $l$ and $w$ are both functions of time then the chain-rule and product-rule for functions of time naturally yield: $$ \frac{dA}{dt} = \frac{dl}{dt}w+l\frac{dw}{dt} \qquad (\ \star \ ) .$$ As a good approximation for small durations of time $\triangle t$ we have $\frac{dA}{dt}\triangle t = \triangle A$. This approximation can be explicitly understood as it connects with the limit definition given earlier and I try to use $\triangle t$ to denote a finite change in time. Generally, $\triangle f$ denotes some finite change. On the other hand, if all the variables involved are time-dependent and we consider a process where the duration of time $\triangle t$ is very small then I say as a heuristic we can think of $\triangle t \rightarrow dt$. Moreover, we may multiply by $dt$ to obtain the relation between the infinitesimal changes in $A,l,w$ for my token example: $$ dA = wdl+ldw $$ My interpretation for them goes something like this: when we write $dA = wdl+ldw$ we intend to approximate the change in $A$ as it related to the change and values of $l,w$. This relation is a short-hand for the related-rates equation $\star$.

Then, later in integration applications, I advocate the method once more. I call it the infinitesimal method. I tell them it is a notational convenience which is justified by more cumbersome expressions which make problems harder to work. Yes, we could set up an explicit Riemann sum for area, volume or work problems but it is far more efficient to calculate formally $dA,dV$ or $dW$ and express each in terms of one variable which parametrizes the situation. I include in my notes discussions of how we go back and forth from the infinitesimal to the currently-traditional function-based framework. The technique of $u$-substitution is far easier to grasp in the formal manipulations of the method of differentials. However, if you want to prove anything, the method of differentials can put you in a bad place. That said, I think there is room for both approaches. The question is what is the task at hand? If the task is foundations and geometric intuition then I tend to think differentials are not so helpful. However, if the task is problem-solving, especially involving integration as a continuous sum, then differentials are quite appropriate. The students can understand them as a convenient notation for a limit-based formulation. There is no need to choose, we can have the best of both worlds.

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A faculty member in my department is a former student of Dray's and 2-3 years ago showed me the differentials approach. He and I both use it quite extensively in our first semester calculus courses.

Here's the pros and cons (in no particular order) and comments on them.

Pro:

1) It's generally awesome. Students in my class struggle a lot less with chain rule, implicit differentiation, related rates, optimization, u-substitution, and integration by parts. (I could go on about how awesome it is, but that's not part of the original question.) In fact, the students that struggle the most are the ones who are retaking it (either again in college or just because they took it in high school) and won't embrace the differentials method. They usually end up making what I assume are the same mistakes they made the first time they took the course.

2) In Calculus II, it makes even more sense. Integration techniques make more sense because $dx$ or $du$ are not just marking the end of the integral, they actually mean something. For the geometrical computations you can talk generally and then have students apply it to the given situation. For example, instead of two formulas for volumes by cylindrical shells, I give them one: $\int 2\pi r h dr$. Then it's up to them to figure out what $r$, $h$, and $dr$ represent geometrically. Likewise, it makes introducing parametric and polar curves easier since students already are comfortable with equations that aren't $y=f(x)$. Also, it greatly opens up the applications of integration section. Work = (Force)*(distance) now becomes $dW = s \cdot dF$ (or $dW = F \cdot ds$ if appropriate, "What is the small quantity?", "Which are you assuming to be constant on your slice?"), which is really how advanced physics/engineering classes think about such relations.

Pros/Cons (depending on how you look at it):

1) It makes calculus extremely algebraic to the point that you really don't have to understand much of the concepts going on. As long as you know what algebraic fact you need/are using (e.g. solving for $dA/dt$ or setting $dA=0$), it's just algebra. Of course, a lot of students are weak in their algebra skills...

2) Transition to Calculus II: If they have me again, no big deal. If they have someone else, I would guess they might stutter a bit in the beginning. But I usually use $d/dx$ at least once or twice around the end of the course and only once in the past 3 years have I had a student ask me what that meant. When I explained it meant, "Take a d, then divide by dx" they gave an "oh, duh" response.

3) Students mix up $3x^2$ and $3x^2dx$, i.e. they sometimes will write a function when they mean a differential or vice versa. If they write a differential and mean a function, it usually isn't too big a deal, since if they evaluate the function they then drop the $dx$. If they write a function and needed a differential, then the error is a problem because they typically need to solve for a ratio of differentials or something and they don't have the right pieces to do so. All in all, I see this as a small trade-off.

4) Students will get confused when they look at the book. I had a lot of problems when I first started using differentials because students would look to the book to do the homework. Now I have notes that are NOT my examples from lecture, but the examples in the book re-worked using differentials. I've had far less problems since. I'm hoping in an upcoming sabbatical to rewrite an open-source Calc book to put it into a differentials flavor...

5) Colleagues' opinions. Well, as I mentioned above, maybe I'm lucky that there's two of us out of the 6-7 Calc I instructors we have at our university. The others either think it's fine (as long as they don't have to do it) or are interested in using bits and pieces (but won't commit to going all out). Nobody challenges the rigor of it because, honestly, given our student population, there's little rigor in our Calc I anyway.

6) Tutors. Usually I warn the tutors who haven't had me what they might see when students come in. Our tutors are typically junior or senior math majors. When I show them the differential approach they take to it very quickly (after only a couple of examples) and usually wonder why everybody doesn't do it that way.

So that's my (more than) 2 cents. I could go on and on, but I think that answers most of your questions.

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    $\begingroup$ Nice answer. I'd love to see your open-source book when it's done. $\endgroup$ – Ben Crowell May 12 '14 at 15:00
  • $\begingroup$ That calculations make mathematics "more accessible" has long been recognized, at least since Descartes and certainly by Newton and Leibniz. Yet, limits have now become, even though they cannot be calculated---looking for a Skolem function $\delta = S(\epsilon)$ hardly qualifies, the rock on which the calculus rests ... and beginners founder. Is there really an open-source book in the making? $\endgroup$ – schremmer Apr 6 '15 at 23:49
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A great quote from Dray and Manogue:

. . . many mathematicians think in terms of infinitesimal quantities: apparently, however, real mathematicians would never allow themselves to write down such thinking, at least not in front of the children. —Bill McCallum [16]

I think in this type of discussion it is important to separate carefully between two issues: (a) what is the theoretical justification for using differentials/infinitesimals in the classroom, and (b) what is the actual situation on the ground, i.e. are teachers successful in imparting understanding of basic notions of the calculus such as slopes/derivatives, areas/integrals, seeking minima/maxima, basic theorems such as the extreme value theorem, etc. using such an approach?

As far as (b) is concerned, I can report based on classroom experience with Keisler' book last term that the approach is successful. Not only was it successful but the students petitioned their department to make sure the class is taught exactly the same way next year. The true infinitesimal calculus course has now been taught for 4 years to over 400 freshmen, with testable success; see this article.

As far as (a) is concerned, of course we know today that infinitesimals are consistent. It is even more important to make the point that traditional calculus courses do not typically establish the foundations of the subject such as constructing the real numbers. Some foundational material has to be taken as given, and is taken up again in more advanced analysis or algebra courses. Similarly, in the approach using calculus certain facts need to be taken as given and Keisler does a marvellous job of motivating whatever facts that are taken as given in this approach.

It is worth mentioning the work by Kathlene Sullivan from the mid 1970s who performed a controlled experiment in the Chicago area, with some groups following the traditional approach, and others using Keisler's book. The groups using infinitesimals came away from the course with a slightly better understanding of the basic calculus notions, according to the conclusions of the study.

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    $\begingroup$ You make good points, but it's also important not to give the impression that differentials can only be done rigorously using infinitesimals. $\endgroup$ – Mike Shulman Jul 3 '14 at 4:33
  • $\begingroup$ @MikeShulman, there are other treatments of differentials of course but the only pedagogically useful one at the level of freshman calculus is via infinitesimals. $\endgroup$ – Mikhail Katz Jul 31 '14 at 14:26
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    $\begingroup$ that's a very bold statement! Have you any proof? $\endgroup$ – Mike Shulman Jul 31 '14 at 22:42
  • $\begingroup$ @MikeShulman, differential forms are too advanced for freshman calculus. As far as the approach with first calculating the derivative and then assigning arbitrary real values to $dx$ and $dy$ so as to fit the equation, it seems tautological and hardly explains anything compared to an explanation provided by an infinitesimal microscope figure. $\endgroup$ – Mikhail Katz Aug 3 '14 at 7:30
  • $\begingroup$ well, you've cited two examples, so to complete your proof you need to show that those are the only other two ways to use differentials in teaching calculus. (-: What about "a differential is an expression of the form $f(x)\, dx$"? $\endgroup$ – Mike Shulman Aug 6 '14 at 5:47
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I had the chance to read Dray & Minogue's online stuff shortly before I was first assigned to teach a Calculus course, an Applied Calculus course intended for biology and economics majors. As it was a terminal math course, I felt justified in taking an unusual approach; as it was applied, I didn't worry about rigour (as long as I knew that it could be done). So I used differentials. It seemed to go all right.

That was three years ago. Since then, I've also taught the regular three-semester-equivalent Calculus series for physical science/engineering and mathematics majors. Sometimes I teach the whole sequence in a row, sometimes I only teach Caclulus 3 (multivariable). I have used differentials every time. I particularly love it for multivariable calculus, where I also introduce differential forms.

It is generally not too hard to link this to what's in the book (which is chosen at the departmental level). After all, differentials (and differential forms in multivariable integrals) appear in the text; it's just that they only appear in certain combinations. So I can pick out a subexpression from the expressions in the book and explain that it has a meaning in its own right. Sometimes I write some new exercises, but mostly I just show them another way to do the exercises in the book. Sometimes they do it my way, sometimes the book's way; it's all good.

In the applied course, I do differentials before derivatives, which I do before limits. We're not pretending to define things rigorously, so there is no harm in this, and this seems to be in order of increasing difficulty. I actually introduce rigorous definitions (even ones that are not in the textbook), but at the end, in the context of approximation methods (which is actually how they first appeared historically); I mention that such-and-such actually provides a rigorous definition, which they take as one of my remarks important for their well-rounded liberal education that won't be on the test, and we move on.

In the regular course, I feel obligated to give the rigorous definitions a more prominent place, since they are explicitly part of the syllabus. This means an approach as in Aeryk's answer: limits, then derivatives, then differentials. I pretty much agree with everything that Aeryk has said, right down the line. But I will add that differentials are especially awesome in multivariable calculus, where I can introduce partial derivatives as in David Butler's answer. Often it is easier to work with the differentials directly, and never mind the individual partial derivatives that appear within them.

I must caution against conflating different kinds of differentials and infinitesimals. In particular, Dray & Minogue's approach is NOT the same as Keisler's approach using nonstandard analysis. (Although a nice thing about Keisler's book is that it allows you to do derivatives before limits, rigorously even.) Dray & Minogue's (and hence my) differentials are the linear differentials from differential geometry (and in differential forms), not the nonstandard hyperreal numbers from nonstandard analysis (nor the nilpotent infinitesimals from synthetic differential geometry, for that matter). To distinguish these, consider: if ‘d’ indicates a finitesimal change, then dy/dx = f'(x) is an approximation; if it indicates a nonstandard infinitesimal, then dy/dx = f'(x) is an adequality (equality of standard parts); if it indicates a linear differential, then dy/dx = f'(x) is an equality, period. (All on the assumption that y = f(x), of course.)

I also want to stress the triviality of the Chain Rule with differentials. I don't mean that you can prove it by cancelling du in dy/dx = dy/du du/dx; it is a real theorem that requires a real proof. (I once read a review of Keisler's book that praised Keisler for being able to prove the Chain Rule by a trivial cancellation; I can only assume that the reviewer did not read that part of the book, for Keisler himself was under no such illusions.) But when working with differentials, it becomes a theoretical result, not a tool for practical calculation. The equation dy/dx = dy/du du/dx is a triviality, but it is not the Chain Rule; or rather, it is the Chain Rule only if you change the meaning of the symbols from one place to another, an unfair trick to play on students. The real Chain Rule is d(f(u)) = f'(u) du. It tells you how, if you can differentiate [take the derivative of] a function, you can apply that function to any differentiable expression and differentiate [take the differential of] the resulting expression.

An example will show the power of this approach. Suppose that you establish, say by the usual argument involving sum-angle formulas and special limits, that the derivative of the sin function is the cosine function. Then you apply the Chain Rule —once!— to conclude that d(sin u) = cos u du for any differentiable expression u; this fact is called the Sine Rule. Now when faced with such expressions as sin(3x²), sin(5x - sin x), sin(2x + 3y), etc, you do NOT use the Chain Rule; you use the Sine Rule. So in the end, instead of having a rule for each algebraic operation, a derivative for each special function, and a Chain Rule that requires you to analyse expressions as composites, you have a rule for each algebraic operation and a rule for each special function, and every differentiation is done by working your way from the outside in, applying the appropriate rule to whatever operation comes next in the expression as it is written down. (This is at least part of what Aeryk means by saying that Calculus becomes algebraic.) The Chain Rule is only needed explicitly if you are dealing with a new or unknown function.

As you can see with sin(2x + 3y) above, differentials work seamlessly with any number of variables. Students have done Algebra with multiple variables, and they can just as easily do Calculus with multiple variables. Some things, such as optimization, graphing, and integration, are legitimately more complicated with multiple variables, but the basic operation of differentiating [taking the differential of] an expression is not. This is why related rates and implicit differentiation comes so smoothly, but it also means that you can talk about partial derivatives right away too. I make sure to do this, even though it is again an aside that's not on the text, because Calculus 3 is not required for everybody who is going to use partial derivatives, after all.

You can see material for my courses at http://tobybartels.name/courses/. Generally the later material is more polished. All that you really need to look at are the notes at the bottom of each course's main page. You will need a DjVu reader to read some of the older stuff (which has been converted to this smaller file format only).

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    $\begingroup$ You make a good point about avoiding the conflation of differentials in the sense of abstract linearization vs. differentials as numbers in the extended number systems ala Keisler etc. When I think about differentials, I am open to the concept of them as "numbers" and I use that as a heuristic, but I always go back to limits, real numbers, and later the differential or differential forms to think about proofs of theorems. To me, the differential notation is a fortunate accident which is unreasonably sucessful in view of the tapestry of theorems it encodes... with a few isolated failures... $\endgroup$ – James S. Cook Apr 22 '17 at 5:18
  • $\begingroup$ So, while I acknowledge the possibility of building calculus without limits, I don't actually make a practice of it. In my thinking, ultimately, calculus may be thought of without regard to its foundational nuts and bolts. Much like the situation with other concrete models we provide. Complex numbers can be viewed as quotients of polynomials, matrices or vectors in the plane. I suspect it's best to say it "is" none of these things. Rather, complex numbers are objects of the form $a+bi$ where $i^2=-1$. Likewise, perhaps we should think harder about what calculus "is". Just an idea... $\endgroup$ – James S. Cook Apr 22 '17 at 5:22
  • $\begingroup$ Mon chapeau for mentioning adequality! $\endgroup$ – Mikhail Katz Apr 25 '17 at 9:58
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I think the best time to really claim that differentials make sense is during solids of revolution.

Trying to convince students that $\int f(x) \textrm{dx}$ is the sum of infinitely many rectangles only works for the top students. The students don't actually use this knowledge when solving problems this early in calculus, so I think it is best skipped at that point.

However, drawing a thin cylinder and claiming that its volume is $\pi r^2 \textrm{dx}$ when oriented one way and $\pi r^2 \textrm{dy}$ when oriented the other way is golden. Shell method? The thickness of the shell is "a little bit of x" (dx) when oriented one way, and dy the other way. The students get to practice using this while doing problems and so this is (in my opinion) the right time for the concept to be stressed.

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  • $\begingroup$ Arc length is the integral of ds, Work is the integral of F dx, etc. The applications of integration chapter is the right place for this to happen. $\endgroup$ – Chris Cunningham Jul 4 '14 at 14:34
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I run the Maths Learning Centre at my University and I came up with the idea of using differentials on the spot when a student asked me to explain how the chain rule works. The problem is that the chain rule just doesn't relate directly to slopes drawn on a graph, so I needed to come up with a different approach. Following the tendency of the Economics lecturers to use differentials I started there. Later in a revision seminar for economics students, I used it as a way to unify their thinking about ordinary and partial differentiation.

I'm not using pure differentials as described in the article but a bit of a mixture. I've been saying that we are looking for a way to describe how big dy is relative to dx. So I write "dy = (?)*dx". Then I say that the name for the ? is the derivative and it's represented by "dy/dx". Later, with partial derivatives (say with z dependent on x and y) we have dz = (?) dx + (??) dy and the name for (?) is $\partial z/\partial x$ and the name for (??) is $\partial z / \partial y$.

I did make it clear that dy/dx is ALSO the formula for a function in its own right, which is where I introduce the f' notation, saying it's another function related to f.

Finally, this is all connected to a pictorial approach that is not the graph of the function. I draw a horizontal number line for x and a separate horizontal number line for y. Then I talk about how when you start at a point on the x-line, the function produces a matching point on the y-line. I draw a few pairs of matching points. Then I pick one of them and draw a little arrow to represent dx. Then I draw a matching little (but slightly bigger) arrow for dy on the second line. And I point to these two arrows and discuss how big one is relative to the other.

For the two-variable function, I draw the x-line and the y-line next to each other, and below in the middle I draw the z-line. Then the dx arrow and the dy arrow each produce a separate arrow on the z-line, which are arranged head-to-tail to make the whole dz arrow. relationship between dx, dy and dz

The students I have used this with one-on-one have seemed to really like this, especially for the two-variable case.

In the future I would like to actually make a table of x's, y's, dx's and dy's so that they can see how the relationship between dx and dy can be different for different x's and therefore be a formula rather than a single number.

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    $\begingroup$ The chain rule does relate directly to slopes drawn on the graph. What else could it relate to? $\endgroup$ – Steven Gubkin Oct 2 '14 at 20:26
  • $\begingroup$ Can you show me please? I would really like to see how the three slopes are geometrically related on a graph, because it really is not obvious to me. $\endgroup$ – DavidButlerUofA Oct 2 '14 at 21:49
  • $\begingroup$ To clarify: I have seen descriptions where you have the outside function graphed, and the composed function graphed, and you see how an adjustment of the x-axis described by the inside function changes the slope of the graph. What's missing is a graph with the slope of the inside function. This is what I mean by it not directly relating to slopes on a graph. One of them seems to be missing! $\endgroup$ – DavidButlerUofA Oct 2 '14 at 22:00
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    $\begingroup$ I cannot produce a picture currently, but I will describe one. We attempt to determine the derivative of $(g \circ f)$ at $x=a$. To do so introduce a 3 dimensional coordinate system $x,y,z$. Graph $f$ in the $xy$ plane, $g$ in the $yz$ plane, and $(g \circ f)$ in the $xz$ plane. Construct the graph of $g \circ f$ by first bouncing from $x$ to $f(x)$, then from $f(x)$ to $g(f(x))$. We want to see what happens at $a$ to the slopes. So increment $a$ by a small amount $\Delta x$. First $f$ stretches this to $f'(a)\Delta x$. Then $g$ stretches that to $g'(f(a))f'(a)\Delta x$. $\endgroup$ – Steven Gubkin Oct 2 '14 at 22:06
  • $\begingroup$ Very nice! But it's still not your ordinary graph is it? It's still at least one step away from your ordinary slope as students usually think of it. That's what I mean by "not directly related". $\endgroup$ – DavidButlerUofA Oct 2 '14 at 22:40

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